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A168684 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^17 = I.

%I A168684 #17 Mar 24 2021 08:05:14
%S A168684 1,7,42,252,1512,9072,54432,326592,1959552,11757312,70543872,
%T A168684 423263232,2539579392,15237476352,91424858112,548549148672,
%U A168684 3291294892032,19747769352171,118486616112900,710919696676665,4265518180055580
%N A168684 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^17 = I.
%C A168684 The initial terms coincide with those of A003949, although the two sequences are eventually different.
%C A168684 First disagreement at index 17: a(17) = 19747769352171, A003949(17) = 19747769352192. - _Klaus Brockhaus_, Mar 30 2011
%C A168684 Computed with MAGMA using commands similar to those used to compute A154638.
%H A168684 G. C. Greubel, <a href="/A168684/b168684.txt">Table of n, a(n) for n = 0..500</a>
%H A168684 <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,-15).
%F A168684 G.f.: (t^17 + 2*t^16 + 2*t^15 + 2*t^14 + 2*t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/ (15*t^17 - 5*t^16 - 5*t^15 - 5*t^14 - 5*t^13 - 5*t^12 - 5*t^11 - 5*t^10 - 5*t^9 - 5*t^8 - 5*t^7 - 5*t^6 - 5*t^5 - 5*t^4 - 5*t^3 - 5*t^2 - 5*t + 1).
%F A168684 G.f.: (1+t)*(1-t^17)/(1 - 6*t + 20*t^17 - 15*t^18). - _G. C. Greubel_, Mar 24 2021
%t A168684 CoefficientList[Series[(1+t)*(1-t^17)/(1 -6*t +20*t^17 -15*t^18), {t, 0, 50}], t] (* _G. C. Greubel_, Aug 03 2016; Mar 24 2021 *)
%t A168684 coxG[{17, 15, -5, 40}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Mar 24 2021 *)
%o A168684 (Magma)
%o A168684 R<t>:=PowerSeriesRing(Integers(), 40);
%o A168684 Coefficients(R!( (1+t)*(1-t^17)/(1 -6*t +20*t^17 -15*t^18) )); // _G. C. Greubel_, Mar 24 2021
%o A168684 (Sage)
%o A168684 def A168684_list(prec):
%o A168684     P.<t> = PowerSeriesRing(ZZ, prec)
%o A168684     return P( (1+t)*(1-t^17)/(1 -6*t +20*t^17 -15*t^18) ).list()
%o A168684 A168684_list(40) # _G. C. Greubel_, Mar 24 2021
%Y A168684 Cf. A003949 (g.f.: (1+x)/(1-6*x)).
%K A168684 nonn
%O A168684 0,2
%A A168684 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009